In this section, we propose the PDD-based algorithm [1]}, [2]} to solve the problem in (REF ). To this end, we first introduce a set of auxiliary variables as \(\mathcal {S}=\lbrace \mathbf {T},\mathbf {v},\mathbf {W},\mathbf {a},\bar{\mathbf {W}},\bar{\mathbf {T}},\bar{\mathbf {a}}\rbrace \) , where \(\mathbf {T}=\mathbf {W}^H\hat{\mathbf {H}}^H\) , \(\hat{\mathbf {H}}\triangleq \left[\hat{\mathbf {h}}_{}^{[1] \ T},\hat{\mathbf {h}}_{}^{[2] \ T}\right]^T\) , \(\mathbf {a}=[\alpha _1,\alpha _2]^T\) , \(\bar{\mathbf {W}}=\mathbf {W}\) , \(\bar{\mathbf {T}}=\mathbf {T}\) , and \(\bar{\mathbf {a}}=\mathbf {a}\) . With these definitions, the optimization problem becomes
\(\begin{array}{cl}\max \limits _{\mathcal {S}}& \log _2 \left(1+\frac{\alpha _{1}^{2}|t_{11}|^2}{\sigma _{h}^{2}\left(||\mathbf {w}_{1}||^2+||\mathbf {w}_{2}||^2\right)+\sigma _{n}^2}\right)+\log _2 \left(1+\frac{\alpha _{2}^{2}|t_{22}|^2}{\alpha _{1}^{2}|t_{12}|^2+\sigma _{h}^{2}\left(||\mathbf {w}_{1}||^2+||\mathbf {w}_{2}||^2\right)+\sigma _{n}^2}\right)\\\operatorname{s.t.}&|\theta _i|=1, i=1,...,M,\\&\operatorname{trace}\left(\bar{\mathbf {W}}\bar{\mathbf {W}}^H\right)\le P, \\&\bar{\mathbf {a}}^T\bar{\mathbf {a}}=1,\\&\left|\bar{t}_{11}\right|^2\ge \left|\bar{t}_{22}\right|^2,\\&\mathbf {T}=\mathbf {W}^H\hat{\mathbf {h}}^H,\bar{\mathbf {T}}=\mathbf {T}, \bar{\mathbf {W}}=\mathbf {W},\bar{\mathbf {a}}=\mathbf {a}.\end{array}\) 
